A smooth $\mathbb{Z}^p$-action on a closed orientable manifold M is cohomologically rigid (CR) if the cohomology of the induced $\mathbb{Z}^p$-module on the smooth functions on M is isomorphic to the real cohomology of the torus Tp. We prove some general properties of a smooth $\mathbb{Z}^p$-action whose first cohomology is isomorphic to Rp. When the manifold is a low-dimensional torus Tq, $1 \le q \le 2$, we prove that any minimal (all orbits are dense) smooth $\mathbb{Z}^p$-action on Tq whose first cohomology is isomorphic to Rp is $C^\infty$ conjugate to an affine $\mathbb{Z}^p$-action. As a corollary we show that the CR $\mathbb{Z}^p$-actions on Tq, $1 \le q \le 2$, are smooth conjugations of affine CR actions.